Junior Cycle Mathematics

Transcription

1

2 Contents Page 3 Introduction to junior cycle Page 4 Rationale Page 5 Aim Page 6 Overview: Links Page 9 Overview: Course Page 13 Expectations for Students Learning outcomes Unifying strand Number strand Geometry and trigonometry strand Algebra and functions strand Statistics and probability strand Page 21 Assessment and Reporting Assessment for the Profile of Achievement Rationale for the Classroom-Based Assessments in mathematics Page 25 Appendix A Page 27 Appendix B

3 3 Introduction to Introduction to junior cycle junior cycle Junior cycle education places students at the centre of the educational experience, enabling them to actively participate in their communities and in society and to be resourceful and confident learners in all aspects and stages of their lives. Junior cycle is inclusive of all students and contributes to equality of opportunity, participation and outcome for all. The junior cycle allows students to make a greater connection with learning by focusing on the quality of learning that takes place and by offering experiences that are engaging and enjoyable for them, and relevant to their lives. These experiences are of a high quality, contribute directly to the physical, mental and social wellbeing of learners, and where possible, provide opportunities for them to develop their abilities and talents in the areas of creativity, innovation and enterprise. The learner s junior cycle programme builds on their learning to date and actively supports their progress in learning and in addition, supports them in developing the learning skills that will assist them in meeting the challenges of life beyond school.

4 4 Rationale Rationale This mathematics specification provides students with access to important mathematical ideas to develop the mathematical knowledge and skills that they will draw on in their personal and work lives. This specification also provides students, as lifelong learners, with the basis on which further study and research in mathematics and many other fields are built. Mathematical ideas have evolved across societies and cultures over thousands of years, and are constantly developing. Digital technologies are facilitating this expansion of ideas and provide new tools for mathematical exploration and invention. While the usefulness of mathematics for problem solving is well known, mathematics also has a fundamental role in both enabling and sustaining cultural, social, economic and technological advances and empowering individuals to become critical citizens. The specification is underpinned by the conception of mathematics as an interconnected body of ideas and reasoning processes that students negotiate collaboratively with teachers and their peers and as independent learners. Number, measurement and geometry, statistics and probability are common aspects of most people s mathematical experiences in everyday personal, study and work situations. Equally important are the essential roles that algebra, functions and relations, logic, mathematical structure and working mathematically play in people s understanding of the natural and social worlds, and the interaction between them. The mathematics specification builds on students prior learning and focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, computational thinking and problem solving. These capabilities enable students to respond to familiar and unfamiliar situations by employing mathematics to make informed decisions and solve problems efficiently. The specification supports student learning across the whole educational system by ensuring that the links between the various components of mathematics, as well as the relationship between mathematics and other subjects, are emphasised. is composed of multiple but interrelated and interdependent concepts and structures which students can apply beyond the mathematics classroom. For example, in science, understanding sources of error and their impact on the confidence of conclusions is vital; in geography, interpretation of data underpins the study of human populations and their physical environments; in history, students need to be able to imagine timelines and time frames to reconcile related events; and in English, deriving quantitative, logical and spatial information is an important aspect of making meaning of texts. Thus the understanding of mathematics developed through study at junior cycle can inform and support students learning across the whole educational system.

5 5 Aim Aim The aim of junior cycle mathematics is to provide relevant and challenging opportunities for all students to become mathematically proficient so that they can cope with the mathematical challenges of daily life and enable them to continue their study of mathematics in senior cycle and beyond. In this specification, mathematical proficiency is conceptualised not as a one-dimensional trait but as having five interconnected and interwoven components: conceptual understanding comprehension of mathematical concepts, operations, and relations procedural fluency skill in carrying out procedures flexibly, accurately, efficiently, and appropriately strategic competence ability to formulate, represent, and solve mathematical problems in both familiar and unfamiliar contexts adaptive reasoning capacity for logical thought, reflection, explanation, justification and communication productive disposition habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence, perseverance and one s own efficacy.

6 6 Overview: Links Overview: Links supports a broad range of learning experiences at junior cycle. Table 1 shows how junior cycle mathematics is linked to central features of learning and teaching in junior cycle. Table 1: Links between junior cycle mathematics and the statements of learning STATEMENTS OF LEARNING The statement SOL 1: The student communicates effectively using a variety of means in a range of contexts in L1. Examples of possible relevant learning Students organise, consolidate and communicate numerical and mathematical thinking clearly and coherently to peers, teachers and others verbally, and in written form using diagrams, graphs, tables and mathematical symbols. SOL 14: The student makes informed financial decisions and develops good consumer skills. Students learn to develop their critical thinking and reasoning skills by making value-for-money calculations and judgements which will enable them to make informed financial decisions. SOL 15: The student recognises the potential uses of mathematical knowledge, skills and understanding in all areas of learning. SOL 16: The student describes, illustrates, interprets, predicts and explains patterns and relationships. Students apply their mathematical knowledge and skills to a wide variety of problems across different subjects, including gathering, analysing, and presenting data, and using mathematics to model real-world situations. Students develop techniques to explore and understand patterns and relationships in both mathematical and nonmathematical contexts. SOL 17: The student devises and evaluates strategies for investigating and solving problems using mathematical knowledge, reasoning and skills. Students develop problem-solving strategies through engaging in tasks for which the solution is not immediately obvious. They reflect on their own solution strategies to such tasks and compare them to those of others as part of a collaborative learning cycle. SOL 18: The student observes and evaluates empirical events and processes and draws valid deductions and conclusions. SOL 24: The student uses technology and digital media tools to learn, communicate, work and think collaboratively and creatively in a responsible and ethical manner. Students generate and summarise data, select appropriate graphical or numerical methods to describe it, and draw conclusions from graphical and numerical summaries of the data. As part of their understanding of mathematical proof they come to appreciate the distinction between contingent deductions from particular cases, and deductions which can be proved to be universally true. Students engage with digital technology to analyse and display data numerically and graphically; to display and explore algebraic functions and their graphs; to explore shapes and solids; to investigate geometric results in a dynamic way; and to communicate and collaborate with others.

7 7 Overview: Links Key Skills In addition to their specific content and knowledge, the subjects and short courses of junior cycle provide students with opportunities to develop a range of key skills. There are opportunities to support all key skills in this course but some are particularly significant. The junior cycle curriculum focuses on eight key skills: Figure 1: Key skills of junior cycle Using language Using numbers Listening and expressing myself Performing and presenting Discussing and debating Using digital technology to communicate Developing good relationships and dealing with conflict Co-operating Respecting difference Contributing to making the world a better place Learning with others Working with others through digital technology WORKING WITH OTHERS COMMUNICATING BEING CREATIVE Imagining Exploring options and alternatives Implementing ideas and taking action Learning creatively Stimulating creativity using digital technology Developing my understanding and enjoyment of words and language Reading for enjoyment and with critical understanding Writing for different purposes Expressing ideas clearly and accurately Developing my spoken language Exploring and creating a variety of texts, including multi-modal texts BEING LITERATE KEY SKILLS BEING NUMERATE MANAGING MYSELF MANAGING INFORMATION & THINKING STAYING WELL Expressing ideas mathematically Estimating, predicting and calculating Developing a positive disposition towards investigating, reasoning and problem-solving Seeing patterns, trends and relationships Gathering, interpreting and representing data Using digital technology to develop numeracy skills and understanding Knowing myself Making considered decisions Setting and achieving personal goals Being able to reflect on my own learning Using digital technology to manage myself and my learning Being healthy and physically active Being social Being safe Being spiritual Being confident Being positive about learning Being responsible, safe and ethical in using digital technology Being curious Gathering, recording, organising and evaluating information and data Thinking creatively and critically Reflecting on and evaluating my learning Using digital technology to access, manage and share content

8 8 Overview: Links KEY SKILL ELEMENTS RELATING TO MATHEMATICS The examples below identify some of the elements that are related to learning activities in mathematics. Teachers can also build many of the other elements of key skills into their classroom planning. The eight key skills are set out in detail in Key Skills of. Table 2: Links between junior cycle mathematics and key skills Key skill Key skill element Examples of possible student learning activities Being creative Being literate Exploring options and alternatives Expressing ideas clearly and accurately As students engage in a task for which the solution is not immediately obvious, they ask questions, explore ideas and alternatives, evaluate ideas and actions and take more responsibility for their learning. Students explain their thinking and justify their reasoning, using mathematical terminology appropriately and accurately. Being numerate Using digital technology to develop numeracy skills and understanding Students use digital technology to analyse and display data numerically and graphically; to display and explore algebraic functions and their graphs; to explore shapes and solids; to investigate geometric results in a dynamic way; and to communicate and collaborate with others. Communicating Using numbers Students use numbers to describe or summarise a situation; to support their reasoning and conclusions; and to convey and explain patterns and relationships. Managing information and thinking Thinking creatively and critically Students engage in rich tasks which require them to use their mathematical knowledge and skills in novel ways. Managing myself Being able to reflect on my own learning They reflect on their own approaches to such tasks and compare them to those of others, evaluating the strengths and weaknesses of different possible approaches. Students reflect on which learning activities they find most effective, using this knowledge to help further their learning in mathematics. Staying well Being confident Students enjoy frequent opportunities to experience success in mathematics. They experience a positive approach to learning in which different approaches are valued and they are encouraged to learn from mistakes. Working with others Learning with others Students work on collaborative tasks with peers in which they develop both their mathematical and their interpersonal skills, offering mutual support and feedback throughout the process.

9 9 Overview: Course Overview: Course The specification for junior cycle mathematics focuses on developing students ability to think logically, strategically, critically, and creatively through the Unifying strand and the four contextual strands: Number; Geometry and trigonometry; Algebra and functions; and Statistics and probability. The specification has been designed for a minimum of 240 hours timetabled student engagement across the three years of junior cycle. This is a minimum and schools should be aware that there are students who would benefit from an engagement of more than 240 hours to realise the national improvement targets set out in the Literacy and Numeracy strategy (DES,2011). Figure 2: The structure of the specification for junior cycle mathematics Unifying Strand Number Geometry & trigonometry Algebra & functions Statistics & probability Element: Building blocks Element: Representation Element: Connections Element: Problem solving Element: Generalisation and proof Element: Communication

10 10 Overview: Course Unifying strand This strand permeates all of the contextual strands and is composed of the six elements of the specification, which are shown below. There is no specific content linked to this strand; rather, its learning outcomes underpin the rest of the specification. Each learning outcome in this strand is applicable to all of the activities and content of the other four strands for example, students should be able to draw on all of their mathematical knowledge and skills to solve a problem or to communicate mathematics. Furthermore, the elements of this strand are interdependent, so that students should develop the different skills associated with each element in tandem rather than in isolation for example, engaging in problem-solving can help students improve their understanding of building blocks and their ability to make connections within mathematics. The elements Elements Building blocks Representation Students should understand and recall the concepts that underpin each strand, and be able to carry out the resulting procedures accurately, effectively, and appropriately. Students should be able to represent a mathematical situation in a variety of different ways and translate flexibly between them. Connections Students should be able to make connections within strands and between strands, as well as connections between mathematics and the real world. Problem solving Generalisation and proof Communication Students should be able to investigate patterns, formulate conjectures, and engage in tasks in which the solution is not immediately obvious, in familiar and unfamiliar contexts. Students should be able to move from specific instances to general mathematical statements, and to present and evaluate mathematical arguments and proofs. Students should be able to communicate mathematics effectively in verbal and written form.

11 11 Overview: Course Number This strand focuses on different aspects of number, laying the groundwork for the transition from arithmetic to algebra. Learners explore different representations of numbers and the connections between them, as well as the properties and relationships of binary operations. They investigate number patterns, and use ratio and proportionality to solve a variety of problems in numerous contexts. Learners are expected to be able to use calculators appropriately and accurately, as well as to carry out calculations by hand and mentally. They appreciate when it is appropriate to use estimation and approximation, including to check the reasonableness of results. Geometry and trigonometry This strand focuses on analysing characteristics and properties of two- and three-dimensional geometric shapes. Learners use geometry and trigonometry to model and solve problems involving area, length, volume, and angle measure. They develop mathematical arguments about geometric relationships and explore the concept of formal proof, using deduction to establish the validity of certain geometric conjectures and critiquing the arguments of others. Algebra and functions This strand focuses on representing and analysing patterns and relationships found in numbers. Building on their work in the Number strand, learners generalise their observations, expressing, interpreting, and justifying general mathematical statements in words and in symbolic notation. They use the idea of equality to form and interpret equations, and the syntactic rules of algebra to transform expressions and solve equations. Learners explore and analyse the relationships between tables, diagrams, graphs, words, and algebraic expressions as representations of functions. Statistics and probability This strand focuses on determining probability from random events and generating and investigating data. Students explore the relationship between experimental and theoretical probability as well as completing a data investigation; from formulating a question and designing the investigation through to interpreting their results in context and communicating their findings. Learners use graphical and numerical tools, including summary statistics and the concepts and processes of probability, to explore and analyse patterns in data. Through these activities, learners gain an understanding of data anaysis as a tool for learning about the world.

12 12 Overview: Course Progression from early childhood to senior cycle EARLY CHILDHOOD Aistear, the early childhood curriculum framework, celebrates early childhood as a time of wellbeing and enjoyment where children learn from experiences as they unfold. Children s interests and play should be the source of their first mathematical experiences. These experiences can become mathematical as they are represented and explored. Young children represent their ideas by talking, but also through models and graphics. From the motoric and singsong beginnings of rhymes and geometric patterns built from unit blocks stem the gradual generalisation and abstraction of patterns throughout the child s day. PRIMARY SCHOOL The mathematics curriculum at primary school aims to provide children with a language and a system through which to analyse, describe, illustrate and explain a wide range of experiences, make predictions, and solve problems. education seeks to enable learners to think and communicate quantitatively and spatially, solve problems, recognise situations where mathematics can be applied, and use appropriate technology to support such applications. The junior cycle mathematics specification consolidates and develops students learning from primary school and as such experience of the learning outcomes in the Primary School Curriculum is assumed. SENIOR CYCLE The junior cycle mathematics specification is developed to align with Leaving Certificate to allow for the effective transfer of knowledge, understanding, and skills from junior to senior cycle. While certain aspects of the strands have been adapted to specifically suit junior cycle for example, having four rather than five strands it is nonetheless clear from the structure of this specification how students learning in junior cycle mathematics should be developed in senior cycle. A good understanding of the knowledge and skills outlined in this specification will lay the foundations for successful engagement with senior cycle mathematics.

13 13 Expectations Expectations for Students for Students Expectations for students is an umbrella term that links learning outcomes with annotated examples of student work in the subject specification. When teachers, students or parents looking at the online specification scroll over the learning outcomes, a link will sometimes be available to examples of work associated with a specific learning outcome or with a group of learning outcomes. The examples of student work will have been selected to illustrate expectations and will have been annotated by teachers and will be made available alongside this specification. The examples will include work that is: exceptional above expectations in line with expectations. The purpose of the examples of student work is to show the extent to which the learning outcomes are being realised in actual cases. Learning outcomes Learning outcomes are statements that describe what knowledge, understanding, skills and values students should be able to demonstrate having studied mathematics in junior cycle. Junior cycle mathematics is offered at Ordinary and Higher level. The majority of the learning outcomes set out in the following tables apply to all students. Additional learning outcomes for those students who take the Higher-level mathematics examination are highlighted in bold. As set out here the learning outcomes represent outcomes for students at the end of their three years of study. The specification stresses that the learning outcomes are for three years and therefore the learning outcomes focused on at a point in time will not have been completed, but will continue to support students learning of mathematics up to the end of junior cycle. The outcomes are numbered within each strand. The numbering is intended to support teacher planning in the first instance and does not imply any hierarchy of importance across the outcomes themselves. The examples of student work linked to learning outcomes will offer commentary and insights that support different standards of student work.

14 14 Unifying strand Elements Students should be able to: Expectations for Students Building blocks U.1 recall and demonstrate understanding of the fundamental concepts and procedures that underpin each strand U.2 apply the procedures associated with each strand accurately, effectively, and appropriately U.3. recognise that equality is a relationship in which two mathematical expressions have the same value Representation U.4 represent a mathematical situation in a variety of different ways, including: numerically, algebraically, graphically, physically, in words; and to interpret, analyse, and compare such representations Connections U.5 make connections within and between strands U.6 make connections between mathematics and the real world Problem solving U.7 make sense of a given problem, and if necessary mathematise a situation U.8 apply their knowledge and skills to solve a problem, including decomposing it into manageable parts and/or simplifying it using appropriate assumptions U.9 interpret their solution to a problem in terms of the original question U.10 evaluate different possible solutions to a problem, including evaluating the reasonableness of the solutions, and exploring possible improvements and/or limitations of the solutions (if any) Generalisation and proof Communication U.11 generate general mathematical statements or conjectures based on specific instances U.12 generate and evaluate mathematical arguments and proofs U.13 communicate mathematics effectively: justify their reasoning, interpret their results, explain their conclusions, and use the language and notation of mathematics to express mathematical ideas precisely

15 15 Expectations for Students Number strand Students should be able to: N.1 investigate the representation of numbers and arithmetic operations so that they can: a. represent the operations of addition, subtraction, multiplication, and division in N, Z, and Q using models including the number line, decomposition, and accumulating groups of equal size b. perform the operations of addition, subtraction, multiplication, and division and understand the relationship between these operations and the properties: commutative, associative and distributive in N, Z, and Q and in R\Q, including operating on surds c. explore numbers written as a b (in index form) so that they can: I. flexibly translate between whole numbers and index representation of numbers II. use and apply generalisations such as a p a q = a p+q ; (a p )/(a q ) = a p q ; (a p ) q = a pq ; and n 1/2 = n, for a Z, and p, q, p q, n N and for a, b, n R, and p, q Q III. use and apply generalisations such as a 0 = 1 ; a p/q = q a p = ( q a) p ; a r = 1/(a r ); (ab) r = a r b r ; and (a/b) r = (a r )/(b r ), for a, b R ; p, q Z; and r Q IV. generalise numerical relationships involving operations involving numbers written in index form V. correctly use the order of arithmetic and index operations including the use of brackets d. calculate and interpret factors (including the highest common factor), multiples (including the lowest common multiple), and prime numbers e. present numerical answers to the degree of accuracy specified, for example, correct to the nearest hundred, to two decimal places, or to three significant figures f. convert the number p in decimal form to the form a 10 n, where 1 a < 10, n Z, p Q, and p 1 and 0 < p < 1 N.2 investigate equivalent representations of rational numbers so that they can: a. flexibly convert between fractions, decimals, and percentages b. use and understand ratio and proportion c. solve money-related problems including those involving bills, VAT, profit or loss, % profit or loss (on the cost price), cost price, selling price, compound interest for not more than 3 years, income tax (standard rate only), net pay (including other deductions of specified amounts), value for money calculations and judgements, mark up (profit as a % of cost price), margin (profit as a % of selling price), compound interest, income tax and net pay (including other deductions)

16 16 Expectations for Students N.3 investigate situations involving proportionality so that they can: a. use absolute and relative comparison where appropriate b. solve problems involving proportionality including those involving currency conversion and those involving average speed, distance, and time N.4 analyse numerical patterns in different ways, including making out tables and graphs, and continue such patterns N.5 explore the concept of a set so that they can: a. understand the concept of a set as a well-defined collection of elements, and that set equality is a relationship where two sets have the same elements b. define sets by listing their elements, if finite (including in a 2-set or 3-set Venn diagram), or by generating rules that define them c. use and understand suitable set notation and terminology, including null set, Ø, subset,, complement, element,, universal set, cardinal number, #, intersection,, union,, set difference, \, N, Z, Q, R, and R\Q d. perform the operations of intersection and union on 2 sets and on 3 sets, set difference, and complement, including the use of brackets to define the order of operations e. investigate whether the set operations of intersection, union, and difference are commutative and/or associative

17 17 Expectations for Students Geometry and trigonometry strand Students should be able to: GT.1 calculate, interpret, and apply units of measure and time GT.2 investigate 2D shapes and 3D solids so that they can: a. draw and interpret scaled diagrams b. draw and interpret nets of rectangular solids, prisms (polygonal bases), cylinders c. find the perimeter and area of plane figures made from combinations of discs, triangles, and rectangles, including relevant operations involving pi d. find the volume of rectangular solids, cylinders, triangular-based prisms, spheres, and combinations of these, including relevant operations involving pi e. find the surface area and curved surface area (as appropriate) of rectangular solids, cylinders, triangular-based prisms, spheres, and combinations of these GT.3 investigate the concept of proof through their engagement with geometry so that they can: a. perform constructions 1 to 15 in Post-Primary School (constructions 3 and 7 at HL only) b. recall and use the concepts, axioms, theorems, corollaries and converses, specified in Post-Primary School (section 9 for OL and section 10 for HL) I. axioms 1, 2, 3, 4 and 5 II. III. theorems 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 15 and 11, 12, 19, and appropriate converses, including relevant operations involving square roots corollaries 3, 4 and 1, 2, 5 and appropriate converses c. use and explain the terms: theorem, proof, axiom, corollary, converse, and implies d. create and evaluate proofs of geometrical propositions e. display understanding of the proofs of theorems 1, 2, 3, 4, 5, 6, 9, 10, 14, 15, and 13, 19; and of corollaries 3, 4, and 1, 2, 5 (full formal proofs are not examinable) GT.4 evaluate and use trigonometric ratios (sin, cos, and tan, defined in terms of right-angled triangles) and their inverses, involving angles between 0 and 90 at integer values and in decimal form GT.5 investigate properties of points, lines and line segments in the co-ordinate plane so that they can: a. find and interpret: distance, midpoint, slope, point of intersection, and slopes of parallel and perpendicular lines b. draw graphs of line segments and interpret such graphs in context, including discussing the rate of change (slope) and the y intercept c. find and interpret the equation of a line in the form y = mx + c; y y 1 = m(x x 1 ); and ax + by + c = 0 (for a, b, c, m, x 1, y 1 Q); including finding the slope, the y intercept, and other points on the line GT.6 investigate transformations of simple objects so that they can: a. recognise and draw the image of points and objects under translation, central symmetry, axial symmetry, and rotation b. draw the axes of symmetry in shapes

18 18 Expectations for Students Algebra and functions strand Students should be able to: AF.1 investigate patterns and relationships (linear, quadratic, doubling and tripling) in number, spatial patterns and real-world phenomena involving change so that they can: a. represent these patterns and relationships in tables and graphs b. generate a generalised expression for linear and quadratic patterns in words and algebraic expressions and fluently convert between each representation c. categorise patterns as linear, non-linear, quadratic, and exponential (doubling and tripling) using their defining characteristics as they appear in the different representations AF.2 investigate situations in which letters stand for quantities that are variable so that they can: a. generate and interpret expressions in which letters stand for numbers b. find the value of expressions given the value of the variables c. use the concept of equality to generate and interpret equations AF.3 apply the properties of arithmetic operations and factorisation to generate equivalent expressions so that they can develop and use appropriate strategies to: a. add, subtract and simplify I. linear expressions in one or more variables with coefficients in Q II. quadratic expressions in one variable with coefficients in Z III. expressions of the form a, where a, b, c Z (bx + c) b. multiply expressions of the form I. a (bx +cy + d); a (bx 2 + cx + d); and ax (bx 2 + cx + d), where a, b, c, d Z II. (ax + b) (cx + d) and (ax + b) (cx 2 + dx + e), where a, b, c, d, e Z c. divide quadratic and cubic expressions by linear expressions, where all coefficients are integers and there is no remainder d. flexibly convert between the factorised and expanded forms of algebraic expressions of the form: I. axy, where a Z II. III. IV. axy + byz, where a, b Z sx ty + tx sy, where s, t Z dx 2 + bx; x 2 + bx + c; and ax 2 + bx + c, where b, c, d Z and a N V. x 2 a 2 and a 2 x 2 b 2 y 2, where a, b Z

19 19 Expectations for Students AF.4 select and use suitable strategies (graphic, numeric, algebraic, trial and improvement, working backwards) for finding solutions to: a. linear equations in one variable with coefficients in Q and solutions in Z or in Q b. quadratic equations in one variable with coefficients and solutions in Z or coefficients in Q and solutions in R c. simultaneous linear equations in two variables with coefficients and solutions in Z or in Q d. linear inequalities in one variable of the form g(x) < k, and graph the solution sets on the number line for x N, Z, and R AF.5 generate quadratic equations given integer roots AF.6 apply the relationship between operations and an understanding of the order of operations including brackets and exponents to change the subject of a formula AF.7 investigate functions so that they can: a. demonstrate understanding of the concept of a function b. represent and interpret functions in different ways graphically (for x N, Z, and R, [continuous functions only], as appropriate), diagrammatically, in words, and algebraically using the language and notation of functions (domain, range, co-domain, f(x) =, f :x, and y =) (drawing the graph of a function given its algebraic expression is limited to linear and quadratic functions at OL) c. use graphical methods to find and interpret approximate solutions of equations such as f(x) = g(x) and approximate solution sets of inequalities such as f(x) < g(x) d. make connections between the shape of a graph and the story of a phenomenon, including identifying and interpreting maximum and minimum points

20 20 Expectations for Students Statistics and probability strand Students should be able to: SP.1 investigate the outcomes of experiments so that they can: a. generate a sample space for an experiment in a systematic way, including tree diagrams for successive events and two-way tables for independent events b. use the fundamental principle of counting to solve authentic problems SP.2 investigate random events so that they can: a. demonstrate understanding that probability is a measure on a scale of 0-1 of how likely an event (including an everyday event) is to occur b. use the principle that, in the case of equally likely outcomes, the probability of an event is given by the number of outcomes of interest divided by the total number of outcomes c. use relative frequency as an estimate of the probability of an event, given experimental data, and recognise that increasing the number of times an experiment is repeated generally leads to progressively better estimates of its theoretical probability SP.3 carry out a statistical investigation which includes the ability to: a. generate a statistical question b. plan and implement a method to generate and/or source unbiased, representative data, and present this data in a frequency table c. classify data (categorical, numerical) d. select, draw and interpret appropriate graphical displays of univariate data, including pie charts, bar charts, line plots, histograms (equal intervals), ordered stem and leaf plots, and ordered back-to-back stem and leaf plots e. select, calculate and interpret appropriate summary statistics to describe aspects of univariate data. Central tendency: mean (including of a grouped frequency distribution), median, mode. Variability: range f. evaluate the effectiveness of different graphical displays in representing data g. discuss misconceptions and misuses of statistics h. discuss the assumptions and limitations of conclusions drawn from sample data or graphical/numerical summaries of data

21 21 Assessment Assessment and Reporting and Reporting Assessment in education involves gathering, interpreting and using information about the processes and outcomes of learning. It takes different forms and can be used in a variety of ways, such as to record and report achievement, to determine appropriate routes for learners to take through a differentiated curriculum, or to identify specific areas of difficulty or strength for a given learner. While different techniques may be employed for formative, diagnostic and summative purposes, the focus of the assessment and reporting is on the improvement of student learning. To do this it must fully reflect the aim of the curriculum. The junior cycle places a strong emphasis on assessment as part of the learning process. This approach requires a more varied approach to assessment in ensuring that the assessment method or methods chosen are fit for purpose, timely and relevant to students. Assessment in junior cycle mathematics will optimise the opportunity for students to become reflective and active participants in their learning and for teachers to support this. This rests upon the provision for learners of opportunities to negotiate success criteria against which the quality of their work can be judged by peer, self, and teacher assessment; and upon the quality of the focused feedback they get in support of their learning. Providing focused feedback to students on their learning is a critical component of high-quality assessment and a key factor in building students capacity to manage their own learning and their motivation to stick with a complex task or problem. Assessment is most effective when it moves beyond marks and grades, and reporting focuses not just on how the student has done in the past but on the next steps for further learning. This approach will ensure that assessment takes place as close as possible to the point of learning. Summative assessment still has a role to play, but is only one element of a broader approach to assessment. Essentially, the purpose of assessment and reporting at this stage of education is to support learning. Parents/guardians should receive a comprehensive picture of student learning. Linking classroom assessment and other assessment with a new system of reporting that culminates in the awarding of the Profile of Achievement (JCPA) will offer parents/guardians a clear and broad picture of their child s learning journey over the three years of junior cycle. To support this, teachers and schools will have access to an Assessment Toolkit. Along with the guide to the Subject Learning and Assessment Review (SLAR) process, the Assessment Toolkit will include learning, teaching and assessment support material, including: formative assessment planning for and designing assessment ongoing assessments for classroom use judging student work looking at expectations for students and features of quality reporting to parents and students thinking about assessment: ideas, research and reflections a glossary.

22 22 Assessment and Reporting The contents of the Assessment Toolkit will include a range of assessment supports, advice and guidelines that will enable schools and teachers to engage with the new assessment system and reporting arrangements in an informed way, with confidence and clarity. Assessment for the Profile of Achievement The assessment of mathematics for the purposes of the Profile of Achievement (JCPA) will comprise two Classroom-Based Assessments: CBA 1; and CBA 2. In addition, the second Classroom-Based Assessment will have a written Assessment Task that will be marked, along with a final examination, by the State Examinations Commission. Rationale for the Classroom-Based Assessments in Over the three years of junior cycle, students will be provided with many opportunities to enjoy and learn mathematics. The Classroom-Based Assessments, outlined below, link to the priorities for learning and teaching in mathematics, with a particular emphasis on problem solving and communicating. Through the Classroom-Based Assessments students will develop and demonstrate their mathematical proficiency by actively engaging in practical and authentic learning experiences. The Classroom-Based Assessments will be carried out by all students, and will be marked at a common level. The teacher s judgement of their mathematical attainment will be recorded for subject learning and assessment review, as well as for the school s reporting to parents and students. Classroom-Based Assessment 1: CBA Format Student preparation Completion of assessment SLAR meeting Mathematical investigation Report which may be presented in a wide range of formats Students will, over a three-week period, follow the Problem-solving cycle to investigate a mathematical problem. Problem-solving cycle: define a problem; decompose it into manageable parts and/or simplify it using appropriate assumptions; translate the problem to mathematics if necessary; engage with the problem and solve it if possible; interpret any findings in the context of the original problem. End of second year One review meeting

23 23 Assessment and Reporting Classroom-Based Assessment 2: CBA Format Student preparation Completion of assessment Statistical investigation Report which may be presented in a wide range of formats Students will, over a three-week period; follow the Statistical enquiry cycle. Statistical enquiry cycle: formulate a question; plan and collect unbiased, representative data; organise and manage the data; explore and analyse the data using appropriate displays and numerical summaries and answer the original question giving reasons based on the analysis section. End of first term of third year SLAR meeting One review meeting Assessing the Classroom-Based Assessments More detailed material on assessment for reporting in junior cycle mathematics, setting out details of the practical arrangements related to assessment of the Classroom-Based Assessments, will be available in separate Assessment Guidelines for. This will include, for example, the suggested length and formats for student pieces of work, and support in using on balance judgement in relation to the features of quality. The NCCA s Assessment Toolkit will also include substantial resource material for use and reference in the ongoing classroom assessment of junior cycle mathematics, as well as providing a detailed account of the Subject Learning and Assessment Review process. Features of quality The features of quality support student and teacher judgement of the Classroom-Based Assessments and are the criteria that will be used by teachers to assess the pieces of student work. All students will complete both CBAs. The features of quality will be available in Assessment Guidelines for.

24 24 Assessment and Reporting Assessment Task The Assessment Task is a written task completed by students during class time, which is not marked by the class teacher, but is sent to the State Examinations Commission for marking. It will be allocated 10% of the marks used to determine the grade awarded by the SEC. The Assessment Task is specified by the NCCA and is related to the learning outcomes on which the second Classroom-Based Assessment is based. The content and format of the Assessment Task may vary from year to year. FINAL EXAMINATION There will be two examination papers, one at Ordinary and one at Higher level, set and marked by the State Examinations Commission (SEC). The examination will be two hours in duration and will take place in June of third year. The number of questions on the examination papers may vary from year to year. In any year, the learning outcomes to be assessed will constitute a sample of the relevant outcomes from the tables of learning outcomes. INCLUSIVE ASSESSMENT PRACTICES This specification allows for inclusive assessment practices whether as part of ongoing assessment or Classroom-Based Assessments. Where a school judges that a student has a specific physical or learning difficulty, reasonable accommodations may be put in place to remove, as far as possible, the impact of the disability on the student s performance in Classroom-Based Assessments. The accommodations, e.g. the support provided by a special needs assistant or the support of assistive technologies, should be in line with the arrangements the school has put in place to support the student s learning throughout the year.

25 25 Appendix A: Glossary Appendix A: Glossary of action verbs of action verbs This glossary is designed to clarify the learning outcomes. Each action verb is described in terms of what the learner should be able to do once they have achieved the learning outcome. This glossary will be aligned with the command words used in the assessment. Action verbs Analyse Apply Calculate Classify Compare Construct Convert Define Discuss Estimate Evaluate Explain Generalise Generate Interpret Investigate Justify Mathematise Students should be able to study or examine something in detail, break down to bring out the essential elements or structure; identify parts and relationships, and to interpret information to reach conclusions select and use knowledge and/or skills to solve a problem in a new situation work out a numerical answer group things based on common characteristics give an account of the similarities and (or) differences between two (or more) items or situations, referring to both (all) of them throughout use properties of shapes and geometric results to draw accurately, using only the prescribed geometrical tools change from one form to another [a set]: give a rule that identifies the elements of a set offer a considered, balanced review that includes a range of arguments, factors or hypotheses; opinions or conclusions should be presented clearly and supported by appropriate evidence state or calculate a rough value for a particular quantity judge the relative quality or validity of something, which may include analysing, comparing and contrasting, criticising, defending, or judging give a reasoned account, showing how causes lead to outcomes generate a general statement based on specific instances produce or create use knowledge and understanding to explain the meaning of something in context observe, study, or make a detailed and systematic examination to establish facts and reach new conclusions give valid reasons or evidence to support an answer or conclusion generate a mathematical representation (e.g. graph, equation, geometric figure) to describe a particular aspect of a phenomenon

26 26 Appendix A: Glossary of action verbs Action verbs Prove Round Sketch Solve State Understand Use Verify Students should be able to give a deductive argument to demonstrate that a particular statement is true, including reasons for each step in the argument give the number in the required form (for example, a multiple of 100, or a number with three significant figures) that is closest in absolute terms to a particular number draw a rough diagram or graph without using geometrical tools work out an answer or solution to provide a concise statement with little or no supporting argument have detailed knowledge of, be able to use appropriately, and see the connections between parts apply knowledge or rules to put theory into practice demonstrate that a statement is true

27 27 School School At a glance: Definitions, Axioms, Theorems and Corollaries Page Number 36 Axiom1. The two points axiom. Definition 1. Segment [AB]. Ray [AB. 37 Definition 2. Collinear. Definition 3. Triangle ABC, side, vertex Definition 4. Distance AB. Length Axiom 2. Ruler axiom. 38 Definition 5. Midpoint. Definition 6. Convex subset of the plane. Vertex, arms and inside of an angle. Definition 7. Null angle. Definition 8. Ordinary angle. Definition 9. Straight angle. Definition 10. Reflex angle. Definition 11. Full angle. 39 Definition 12. Angle notation BAC Axiom 3. Protractor Axiom. Definition 13. Bisector of an angle. Definition 14. Right angle. 40 Definition 15. Acute angle. Definition 16. Supplementary angles. Definition 17. Perpendicular lines. Definition 18. Vertically opposite angles. Theorem 1. Vertically Opposite angles are equal in measure. Definition 19. Congruent triangles.

28 28 School Page Number 41 Axiom 4. Congruent triangles. Definition 20. Right angled triangle. Hypotenuse Definition 21. Isosceles triangle. Equilateral. Scalene Theorem 2. In an isosceles triangle, the angles opposite the equal sides are equal. Converse Theorem 2. If two angles are equal, then the tringle is isosceles. 42 Definition 22. Parallel lines. Axiom 5. Axiom of parallels. Definition 23. Transversal line. Definition 24. Alternate angles. 43 Theorem 3. If a transversal makes equal alternate angles on two lines, then the lines are parallel. Converse of Theorem 3. If two lines are parallel, then any transversal will make equal alternate angles with them. 44 Theorem 4. The angles in any triangle add to 180 degrees. 45 Definition 25. Corresponding angles. Theorem 5. Two lines are parallel if and only if for any transversal, corresponding angles are equal. 46 Definition 26. Exterior angle. Interior opposite angles. Theorem 6. Each exterior angle of a triangle is equal to the sum of the interior opposite angles. 47 Theorem 7. In a triangle, the angle opposite the greater of two sides is greater than the angle opposite the lesser side. Converse of Theorem 7. The side opposite the greater of two angles is greater than the side opposite the lesser angle 48 Theorem 8. Two sides of a triangle are together greater than the third. 49 Definition 27. Perpendicular bisector. Definition 28. Polygon, sides, vertices, adjacent sides, adjacent vertices, adjacent angles. Definition 29. Quadrilateral, opposite side, opposite angles. 50 Definition 30. Rectangle. Definition 31. Rhombus. Definition 32. Square. Definition 33. Polygon, equilateral and regular. Definition 34. Parallelogram. Theorem 9. In a parallelogram, opposite sides are equal, and opposite angles are equal.